Chemistry Reference
In-Depth Information
are equal for the doubly occupied MO
and are usually denoted by R in MO theory. We then obtain for the
electron and spin densities of our bond orbital
and
The coefficients of
aa
bb
r
Þþr
1
ð
r
r
Þ¼r
1
ð
r
P
ð
r
;
;
;
r
Þ¼
2R
ð
r
;
r
Þ¼
2
fð
r
Þf
ð
r
Þ
ð
6
:
40
Þ
r
Þr
1
ð
r
r
Þ¼r
1
ð
r
Q
ð
r
;
;
;
r
Þ¼
0
ð
6
:
41
Þ
as expected for an MO doubly occupied by electrons with opposite spin
and M
S
¼
0.
The electron density (6.40) can be further analysed in terms of elemen-
tary contributions from the AOs, giving the so-called population analysis,
which shows how the electrons are distributed between the different
atomic orbitals in the molecule. We obtain
B
ð
r
Þþ
q
AB
x
A
ð
r
Þx
B
ð
r
Þ
þ
q
BA
x
B
ð
r
Þx
A
ð
r
Þ
S
2
A
ð
r
Þþ
q
B
x
2
P
ð
r
;
r
Þ¼
q
A
x
ð
6
:
42
Þ
S
2
2
from (6.36), where
B
ð
r
Þ
are the atomic densities and
ðx
A
ð
r
Þx
B
ð
r
ÞÞ=
S and
ðx
B
ð
r
Þx
A
ð
r
ÞÞ=
S are the overlap densities, all normal-
ized to 1, while the coefficients
x
A
ð
r
Þ
and
x
2
2
2
l
q
A
¼
lS
;
q
B
¼
ð
6
:
43
Þ
2
2
1
þ
l
þ
2
1
þ
l
þ
2
lS
are the atomic charges and
2
lS
q
AB
¼
q
BA
¼
ð
6
:
44
Þ
2
1
þ
l
þ
2
lS
are the overlap charges. The charges are normalized so that
2
2
þ
2
l
þ
4
lS
q
A
þ
q
B
þ
q
AB
þ
q
BA
¼
lS
¼
2
ð
6
:
45
Þ
2
1
þ
l
þ
2
the total number of electrons in the bond orbital
fð
r
Þ
.
For a homopolar bond,
l
¼
1:
1
1
þ
S
;
S
1
þ
S
q
A
¼
q
B
¼
q
AB
¼
q
BA
¼
ð
6
:
46
Þ
